Term Rewriting System R:
[x, y, z]
implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))
IMPLIES(x, or(y, z)) -> IMPLIES(x, z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

IMPLIES(x, or(y, z)) -> IMPLIES(x, z)
IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))


Rules:


implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

IMPLIES(x, or(y, z)) -> IMPLIES(x, z)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
IMPLIES(x1, x2) -> x2
or(x1, x2) -> or(x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pair:

IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))


Rules:


implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))
one new Dependency Pair is created:

IMPLIES(not(x0), or(not(x''), z'')) -> IMPLIES(not(x''), or(x0, z''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

IMPLIES(not(x0), or(not(x''), z'')) -> IMPLIES(not(x''), or(x0, z''))


Rules:


implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes